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A087981 E.g.f.: exp(-2*x) / (1-x)^2. 18
1, 0, 2, 4, 24, 128, 880, 6816, 60032, 589312, 6384384, 75630080, 972387328, 13483769856, 200571078656, 3185540657152, 53800242216960, 962741176500224, 18195808235880448, 362183230599856128, 7572922094360723456, 165945771111208714240, 3802923921298533384192, 90965940197460917878784, 2267151124921333646884864 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,3
COMMENTS
Permanent of an (n+1) X (n+1) (+1, -1)-matrix with exactly n -1's on the diagonal and 1's everywhere else.
It is conjectured by Kräuter and Seifter that for n >= 5 a(n-1) is the maximal possible value for the permanent of a nonsingular n X n (+1, -1)-matrix. I do not know for which values of n this has been confirmed - compare A087982. - N. J. A. Sloane
The Kräuter conjecture on permanents is true (see Budrevich and Guterman). - Sergei Shteiner, Jan 17 2020
The maximal possible value for the permanent of a singular n X n (+1, -1)-matrix is obviously n!.
Degree of the "hyperdeterminant" of a multilinear polynomial on (\P^1(\C))^n, or equivalently of an element of (\C^2)^{⊗ n}: see Gelfand, Kapranov and Zelevinsky. - Eric Rains, Mar 15 2004
(-1)^n * a(n) = Polynomials in A010027 evaluated at -1. - Ralf Stephan, Dec 15 2004
a(n) is the number of n X n (-1, 0, 1)-matrices containing in every row and every column exactly one -1 and one 1 such that the main diagonal does not contain 0's. - Vladimir Shevelev, Apr 01 2010
a(n) is the number of colored permutations with no fixed points of n elements where each cycle is one of two colors. - Michael Somos, Jan 19 2011
Binomial transform is A000255. Hankel transform is A059332. - Paul Barry, Apr 11 2011
Exponential self-convolution of subfactorials (A000166). - Vladimir Reshetnikov, Oct 07 2016
REFERENCES
I. M. Gelfand, M. M. Kapranov, and A. V. Zelevinsky, Discriminants, Resultants and Multidimensional Determinants, Birkhauser, 1994; see Corollary 2.10 in Chapter 14 (p. 457).
LINKS
Mikhail V. Budrevich and Alexander E. Guterman, Kräuter conjecture on permanents is true, arXiv:1810.04439 [math.CO], 2018.
Steven Finch, Rounds, Color, Parity, Squares, arXiv:2111.14487 [math.CO], 2021.
I. M. Gelfand, M. M. Kapranov, and A. V. Zelevinsky, Hyperdeterminants, Advances in Mathematics 96(2) (1992), 226-263; see Corollary 3.10 (p. 246).
Arnold R. Kräuter, Permanenten - Ein kurzer Überblick, Séminaire Lotharingien de Combinatoire, B09b (1983), 34 pp.
Arnold R. Kräuter and Norbert Seifter, Some properties of the permanent of (1,-1)-matrices, Linear and Multilinear Algebra 15 (1984), 207-223.
Norbert Seifter, Upper bounds for permanents of (1,-1)-matrices, Israel J. Math. 48 (1984), 69-78.
Edward Tzu-Hsia Wang, On permanents of (1,-1)-matrices, Israel J. Math. 18 (1974), 353-361.
Wikipedia, Hyperdeterminant
FORMULA
Krauter and Seifter prove that the permanent of an n X n {-1, 1} matrix is divisible by 2^{n - [log_2(n)] - 1}.
Let c(n) be the permanent of the {-1, +1}-matrix of order n X n with n diagonal -1's only. Let a(n) be the permanent of the {-1, +1}-matrix of order (n+1) X (n+1) with n diagonal -1's only. Then by expanding along the first row (like determinant, but with no sign) we get c(n+1) = -c(n) + n a(n-1), a(n) = c(n) + n a(n-1), with c(2) = 2, a(2) = 2. {c(n)} has e.g.f. exp(-2x)/(1-x), see A000023. Also a(n) = c(n+1) + 2*c(n).
The following 4 formulas hold: a(n) = Sum_{k = 0..n} C(n, k)*D_k*D_{n-k}, where D_n = A000166(n); a(n) = n!*Sum_{j = 0..n} (n+1-j)*(-2)^j/j!; a(0) = 1, a(1) = 0 and, for n > 0, a(n+1) = n*(a(n) + 2*a(n-1)); a(0) = 1 and, for n > 0, a(n) = (n*(n+3)/(n+2))*a(n-1) - (-2)^(n+1)/(n+2). - Vladimir Shevelev, Apr 01 2010 [edited by Michael Somos, Jan 19 2011]
G.f.: 1/(1-2x^2/(1-2x-6x^2/(1-4x-12x^2/(1-6x-20x^2/(1-.../(1-2n*x-(n+1)(n+2)x^2/(1-... (continued fraction). - Paul Barry, Apr 11 2011
E.g.f.: 1/U(0) where U(k)= 1 - 2*x/( 1 + x/(2 - x - 4/( 2 + x*(k+1)/U(k+1)))) ; (continued fraction, 3rd kind, 4-step). - Sergei N. Gladkovskii, Oct 28 2012
G.f.: 1/Q(0), where Q(k) = 1 + 2*x - x*(k+2)/(1 - x*(k+1)/Q(k+1)); (continued fraction). - Sergei N. Gladkovskii, Apr 22 2013
G.f.: 1/Q(0) where Q(k) = 1 - 2*k*x - x^2*(k + 1)*(k+2)/Q(k+1); (continued fraction). - Sergei N. Gladkovskii, May 10 2013
G.f.: S(x)/x - 1/x = G(0)/x - 1/x, where S(x) = sum(k >= 0, k!*(x/(1+2*x))^k ), G(k) = 1 + (2*k + 1)*x/( 1+2*x - 2*x*(1+2*x)*(k+1)/(2*x*(k+1) + (1+2*x)/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Jun 26 2013
a(n) = (-2)^n*hypergeom([2, -n], [], 1/2) = 4*(-2)^n*(1 - 2*hypergeom([1, -n-3], [], 1/2))/(n^2+3*n+2) = (4*(-2)^n + Gamma(n+4, -2)*exp(-2))/(n^2+3*n+2). - Vladimir Reshetnikov, Oct 07 2016
a(n) ~ sqrt(2*Pi) * n^(n+3/2) / exp(n+2). - Vaclav Kotesovec, Oct 08 2016
a(n) = KummerU(-n, -n - 1, -2). - Peter Luschny, May 10 2022
EXAMPLE
G.f. = 1 + 2*x^2 + 4*x^3 + 24*x^4 + 128*x^5 + 880*x^6 + 6816*x^7 + ...
Since a(1) = 0, then, for n = 2, we have a(2) = -(-2)^3/4 = 2; further, for n = 3, we find a(3) = (3*6/5)*2 - (-2)^4/5 = 36/5 - 16/5 = 4. - Vladimir Shevelev, Apr 01 2010
a(4) = 24 because there are 6 derangements with one 4-cycle with 2^1 ways to color each derangement and 3 derangements with two 2-cycles with 2^2 ways to color each derangement. - Michael Somos, Jan 19 2011
MAPLE
seq(simplify(KummerU(-n, -n-1, -2)), n = 0..24); # Peter Luschny, May 10 2022
MATHEMATICA
Range[0, 20]! CoefficientList[Series[Exp[-2 x]/(1 - x)^2, {x, 0, 20}], x]
Table[(-2)^n HypergeometricPFQ[{2, -n}, {}, 1/2], {n, 0, 20}] (* Vladimir Reshetnikov, Oct 07 2016 *)
PROG
(PARI) {a(n) = if( n<0, 0, n! * polcoeff( exp( -2 * x + x * O(x^n) ) / ( 1 - x )^2, n ) )} /* Michael Somos, Jan 19 2011 */
CROSSREFS
Sequence in context: A170931 A317999 A164313 * A002875 A330504 A330511
KEYWORD
nonn,easy,nice
AUTHOR
Gordon F. Royle, Oct 28 2003
EXTENSIONS
More terms from Jaap Spies, Oct 28 2003
Further terms from Gordon F. Royle, Oct 29 2003
Definition via e.g.f. from Eric Rains, Mar 15 2004
Changed the offset and terms to correspond to e.g.f, Michael Somos, Jan 19 2011
STATUS
approved

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Last modified March 19 01:57 EDT 2024. Contains 370952 sequences. (Running on oeis4.)